Fuzzy portfolio selection with prospect consistency constraint based on possibility theory

Abstract

Considering that most studies have taken the investors’ preference for risk into account but ignored the investors’ preference for assets, in this paper, we combine the prospect theory and possibility theory to provide investors with a portfolio strategy that meets investors’ preference for assets. Firstly, a novel reference point is proposed to give investors a comprehensive impression of assets. Secondly, the prospect return rate of assets is quantified as trapezoidal fuzzy number, and its possibilistic mean value and variance are regarded as prospect return and risk and then used to define the fuzzy prospect value. This new definition is presented to denote the score of an asset in investors’ subjective cognition. And then, a prospect asset filtering frame is proposed to help investors select assets according to their preference. When assets are selected, another new definition called prospect consistency coefficient is proposed to measure the deviation of a portfolio strategy from investors’ preference. Some properties of the definition are presented by rigorous mathematical proof. Based on the definition and its properties, a possibilistic model is constructed, which can not only provide investors optimal strategies to make profit and reduce risk as much as possible, but also ensure that the deviation between the strategies and investors’ preference is tolerable. Finally, a numerical example is given to validate the proposed method, and the sensitivity analysis of parameters in prospect value function and prospect consistency constraint is conducted to help investors choose appropriate values according to their preferences. The results show that compared with the general M-V model, our model can not only better satisfy investors’ preference for assets, but also disperse risk effectively.

Keywords

Possibility theory prospect theory portfolio selection asset altering framework prospect consistency coefficient

1 Introduction

To help investors disperse risk and make rational investment to some extent, portfolio selection has drawn a great attention from lots of researchers for many years. One of the most famous portfolio selection models is the M-V model (mean-variance probabilistic model), which was originally proposed by Markowitz [1] in 1952 and then became the basis of modern portfolio selection theory. In the M-V model, the distribution of return on an asset is considered as a random variable, and the mean value and variance of the variable are used to represent the return and risk of the asset respectively. M-V model requires investors to provide a given level of risk as the constraint and then maximizes the expected return, or provide a given level of return as the constraint and then minimize the risk. Although the model can balance return and risk to a certain extent, it is rough for the complex financial markets. Thus, many researchers studied the model and made some improvements, including adding complex realistic constraints or adopting more appropriate risk measures.

In recent years, more and more researchers focused on other ways to deal with the uncertainty of financial market. The prevalent view is that there are not only probabilistic factors but also non-probabilistic factors in the market. Therefore, it is necessary to search for an effective way to cope with these non-probabilistic factors. The fuzzy theory proposed by Zadeh [2] is one of the most popular tools to deal with non-probabilistic uncertainty for many fields, such as water quality [3], risk analysis [4], renewable power sources [5] and battery energy storage system [6]. For its effectiveness, many researchers combined it with portfolio selection to construct fuzzy models.

Although the uncertainty of financial market is well dealt with, the problem of investors’ subjective uncertainty is still exists. Most of the portfolio selection models are based on the hypothesis that investors are completely rational and not affected by the historical data, which makes the expected utility theory popular. However, in real-life investment, the investors are always bounded rationality while facing the risk [5], which would make them choose a portfolio that differs from the optimal one. Thus, integrating the irrational behavior of investors into portfolio selection model could provide them with more practical and flexible investment strategies. Prospect theory developed by Kahneman and Tversky [7] has outstanding performance on the decision making problem with bounded rationality, and has been widely applied in multi-attribute decision-making fields [8 –12]. Although some researchers combined prospect theory with portfolio selection [13, 14], the studies in this field are still comparatively insufficient.

In this paper, we aim to combine prospect theory with portfolio selection to provide investors with strategies that satisfy their preference for assets. The main works of this paper are as follows:

To introduce the prospect theory and give investors a comprehensive impression of all assets, a novel reference point is proposed based on the joint possibility distribution of assets.

A new kind of fuzzy prospect value is proposed to denote the relative value of an asset, which can describe the return rate of an asset under unit risk in investors’ subjective cognition. It is used to construct the prospect asset filtering frame to help investors select assets.

To measure the consistency between an optimal strategy provided by portfolio selection model and the investors’ preference for the assets, we define the prospect consistency coefficient and then present some theorems and proofs.

In order to ensure that the optimal strategy can satisfy investors’ preference for assets, a new model with prospect consistency constraint is constructed. The numerical example shows that compared with the general M-V model, our model can not only better satisfy investors’ preference for assets, but also disperse risk effectively.

The rest of this paper is organized as follows: In Section 2, we review some related works of portfolio selection. In Section 3, some basic definitions of possibility theory and prospect theory are introduced. Section 4 presents a novel reference point and the definition of fuzzy prospect value to construct the prospect asset filtering framework. In Section 5, the definition of prospect consistency coefficient and the possibilistic model with prospect consistency constraint are proposed. The numerical example is given in Section 6 and the sensitivity analysis is conducted in Section 7. Finally, the conclusion is given in Section 8.

2 Related work

2.1 Portfolio selection based on probability theory

Most of the portfolio selection models are developed from the M-V model proposed by Markowitz [1]. These models took the complex realities into account and then improved the model. For example, Chunhachinda et al. [15] pointed out that some major stock markets are not normally distributed and the incorporation of skewness into portfolio causes a major change in the construction of the optimal portfolio; Mansini and Speranza [16] dealt with the portfolio problem with minimum transaction lots and showed that the problem of finding a feasible solution is, independently of the risk function, NP-complete in this case; Maccheroni et al. [17] proposed a portfolio selection model based on a class of preferences, and they stressed that these preferences can only ensure consistency with mean-variance preferences in the monotonicity domain; Wang and Liu [18] investigated the multi-period mean-variance portfolio selection problems with fixed and proportional transaction costs; Masmoudi and Abdelaziz [19] studied the portfolio selection problem when the loss in the portfolio return is considered as a recourse cost. For this problem, they found that the investor would penalize unfeasible solutions for uncertain constraints with the most probable highest recourse cost rather than with the expected recourse cost as in the traditional recourse approach.

What these models have in common is that they are based on probability theory. However, the financial market is always too complex to predict the probabilistic distribution of assets. In addition, its uncertainty cannot be completely described by probabilistic factors. Thus, in this paper, we use another effective tool, i.e. fuzzy set, to deal with the complex uncertainty of financial market.

2.2 Portfolio selection based on fuzzy theory

The fuzzy theory proposed by Zadeh [2] provides an effective way to deal with non-probabilistic uncertainty for many fields, and has been widely used in portfolio selection. Based on the possibility theory proposed by Carlsson and Fullér [20], Chen et al. [21] presented a possibilistic mean VaR model for portfolio selection; Deng and Li [22] studied the portfolio selection problem with borrowing constraint; Zhang et al. [23] put forward a fuzzy programming approach for multi-period portfolio optimization with return demand and risk control; Zhou et al. [24] studied stock portfolio selection problem based on varying conservative-neutral-aggressive attitudes where the return rates of stocks are characterized by fuzzy variables; Li and Yi [25] developed a new trapezoidal fuzzy number with an adaptive index to capture the coherence of investor’s expectation; Zhang [26] thought that practical portfolio selection problems often involve the mixture of the stochastic returns with fuzzy information, so that he proposed a mean variance random credibilitic portfolio selection model with different convex transaction costs.

Some of these studies may take the investors’ preference for risk into account, thus they can offer investors more practical strategies. Nevertheless, for all we know, there are few studies that have considered the investors’ preference for assets. In fact, the investors’ preference for assets will affect their acceptance of the strategies provided by the model. In this respect, it is necessary to introduce the investors’ preference for assets into portfolio selection model, which is one of the highlights of our paper.

2.3 Decision-Making based on prospect theory

In portfolio selection, investors may choose investment strategy according to the returns and risks of assets. If we take the returns and risks of assets as two different criteria, it is no doubt that portfolio selection problem can be regarded as a kind of multi-criteria decision-making (MCDM) problem. Most MCDM problems have attached great importance to the psychological factors of decision makers. Yu et al. [27] proposed some new formulae to calculate the gain and loss for an unbalanced HFLTS over another, and then extended the classical TODIM method to deal with multi-criteria group decision making (MCGDM) problems with unbalanced HFLTSs by considering the psychological behavior of decision makers; Zhang et al. [28] presented an extended logarithmic least squares method to derive a priority weight vector from an fuzzy preference relation with self-confidence (FPR-SC), and then applied it to develop a novel approach to two-sided matching decision making with FPR-SC; Zhang et al. [29] defined two consensus measures to identify the leadership of experts and designed a new feedback mechanism to help experts reach consensus in group decision-making process.

Prospect theory, which developed by Kahneman and Tversky [7], focuses on people’s irrational decision-making psychology. It pointed out that people may be influenced by the relative values of alternatives in decision-making process. For its effectiveness, many studies have used prospect theory to solve MCDM problem. Lin et al. [30] set up a dynamic price recommendation method for housing purchases and applied the fuzzy multiple criteria decision making (FMCDM) technique based on the prospect theory of the loss aversion effect; According to the time degree and the prospect value deviation of alternatives in different periods, Dai et al. [31] constructed the triangular fuzzy prospect decision matrices to solve the multi-stage multi-attribute decision-making problem with completely unknown period weights and attribute weights; Tian et al. [32] developed an ULZN-based QUALIFLEX (qualitative flexible multiple criteria method) by considering the decision-maker’s psychological behavior character, and then utilized it to solve a qualitative MCDM problem with larger number of criteria than alternatives; Ma and Zhang [33] proposed a distance measure for normal cloud, set the positive ideal reference point based on prospect theory and then established an optimization model for handling incomplete attribute weights to obtain the ranking of alternatives.

Considering that most studies have taken the investors’ preference for risk into account [13, 14] but ignored the investors’ preference for assets, and the studies of portfolio selection based on prospect theory is insufficient, in this paper, we combine the prospect theory and possibility theory to provide investors with strategies that can meet their preference for assets. Firstly, a novel reference point is proposed to compare the assets with all the others and then provide investors with a comprehensive impression of these assets. Secondly, the prospect return rates of each asset are quantified by the subtraction of trapezoidal fuzzy number to obtain the prospect possibilistic mean values and variances as prospect returns and risks. Based on the prospect return and risk, a new definition called fuzzy prospect value is presented to denote the score of an asset in investors’ subjective cognition. And then, a prospect asset filtering frame is put forward to help investors select a certain number of assets from a large set, which can ensure that all the selected assets have the highest score to satisfy investors’ preference.

In order to provide investors with optimal strategies based on their preference for assets, another new definition called prospect consistency coefficient is proposed. The definition can measure the deviation of a portfolio strategy from investors’ preference with range [0, 1]. And then, a possibilistic model based on the definition is constructed. The model can not only provide investors with optimal strategies to make profit and reduce risk as much as possible, but also ensure that the deviation between the strategies and investors’ preference is tolerable. In addition, based on the Shannon entropy [34], it is proved that our model can effectively disperse risk compared with the general model.

3 Preliminaries

In this section, some definitions of possibility theory and prospect theory will be briefly mentioned for the following sections.

3.1 Possibility theory

A fuzzy number is formally interpreted as a normalized convex fuzzy set with its membership function being piece-wise continuous [25]. The membership function is used to represent the membership degree of the variable belonging to the fuzzy set. Generalized trapezoidal fuzzy number is one of the most typical fuzzy numbers, which is presented as follows:

Definition 1. [35] A fuzzy number R = (r₁, r₂, r₃, r₄) is called a generalized trapezoidal fuzzy number if its membership function μ (x) : x → [0, 1] can be written as: $μ (x) = {\begin{matrix} \frac{x - r_{1}}{r_{2} - r_{1}}, r_{1} ⩽ x ⩽ r_{2}, \\ 1, r_{2} ⩽ x ⩽ r_{3}, \\ \frac{x - r_{4}}{r_{3} - r_{4}}, r_{3} ⩽ x ⩽ r_{4}, \\ 0, otherwise . \end{matrix}$ (1)

If r₂ = r₃, R regresses to a generalized triangular fuzzy number; If r₁ = r₂ and r₃ = r₄, R regresses to an interval number; If r₁ = r₂ = r₃ = r₄, R regresses to a real number.

Definition 2. [12] Let A be a fuzzy number and [A] ^γ be the γ-set of A, where ${[A]}^{γ} = {x \in ℝ | μ_{A} (x) ⩾ γ}$ , $ℝ$ is the real line, μ_A (x) is the membership function of A. According to the definition of fuzzy number, [A] ^γ is a compact subset of $ℝ$ if γ ∈ [0, 1], so that it can be written as [A] ^γ = [a₁ (γ) , a₂ (γ)] (a₁ (γ) ⩽ a₂ (γ)). Then, the possibilistic mean value of A is denoted by: $M (A) = \int_{0}^{1} γ (a_{1} (γ) + a_{2} (γ)) d γ .$ (2)

By formula (2), the possibilistic mean value of generalized trapezoidal fuzzy number R = (r₁, r₂, r₃, r₄) is $M (R) = \frac{1}{3} (r_{2} + r_{3}) + \frac{1}{6} (r_{4} + r_{1}) .$ (3)

Definition 3. [20] Let A be a fuzzy number and [A] ^γ = [a₁ (γ) , a₂ (γ)] (a₁ (γ) ⩽ a₂ (γ)) be the γ-set of A. Then, the possibilistic variance of A is denoted by: $Var (A) = \frac{1}{2} \int_{0}^{1} {γ (a_{2} (γ) - a_{1} (γ))}^{2} d γ .$ (4)

According to formula (4), the possibilistic variance of generalized trapezoidal fuzzy number R = (r₁, r₂, r₃, r₄) is $Var (R) = \frac{1}{12} {(r_{3} - r_{2})}^{2} + \frac{1}{24} {(r_{4} + r_{3} - r_{2} - r_{1})}^{2} .$ (5)

Definition 4. [20] Let A, B be two fuzzy numbers and [A] ^γ = [a₁ (γ) , a₂ (γ)] (a₁ (γ) ⩽ a₂ (γ)), [B] ^γ = [b₁ (γ) , b₂ (γ)] (b₁ (γ) ⩽ b₂ (γ)) be the γ-sets of the fuzzy numbers. Then, the possibilistic covariance of A and B is denoted by: $Cov (A, B) = \frac{1}{2} \int_{0}^{1} γ (a_{2} (γ) - a_{1} (γ)) (b_{2} (γ) - b_{1} (γ)) d γ .$ (6)

According to formula (6), the possibilistic covariance of generalized trapezoidal fuzzy numbers R₁ = (r₁₁, r₁₂, r₁₃, r₁₄) and R₂ = (r₂₁, r₂₂, r₂₃, r₂₄) is

$\begin{matrix} Cov (R_{1}, R_{2}) = \frac{1}{12} (r_{13} - r_{12}) (r_{23} - r_{22}) + \frac{1}{24} \\ (r_{14} + r_{13} - r_{12} - r_{11}) (r_{24} + r_{23} - r_{22} - r_{21}) . \end{matrix}$ (7)

From these definitions, it is not difficult to find that possibilistic mean value, possibilistic variance and possibilistic covariance have the following properties, which are similar to the properties of mean, variance and covariance in probability theory.

Property 1. [20] Let A₁, A₂, ⋯ , A_n be n fuzzy numbers and λ₀, λ₁, λ₂, ⋯ , λ_n be n real numbers. Then, the possibilistic mean value of fuzzy number $A = λ_{0} + \sum_{i = 1}^{n} λ_{i} A_{i}$ is: $M (A) = M (λ_{0} + \sum_{i = 1}^{n} λ_{i} A_{i}) = λ_{0} + \sum_{i = 1}^{n} λ_{i} M (A_{i}) .$ (8)

Property 2. [20] Let A₁, A₂, ⋯ , A_n be n fuzzy numbers and λ₀, λ₁, λ₂, ⋯ , λ_n be n real numbers. Then, the possibilistic variance of fuzzy number $A = λ_{0} + \sum_{i = 1}^{n} λ_{i} A_{i}$ is: $Var (A) = Var (λ_{0} + \sum_{i = 1}^{n} λ_{i} A_{i}) = \sum_{i, j = 1}^{n} λ_{i} λ_{j} Cov (A_{i}, A_{j} .$ (9)

Definition 5. [35] Let A = (a₁, a₂, a₃, a₄), B = (b₁, b₂, b₃, b₄) be two trapezoidal fuzzy numbers, the subtraction rule of trapezoidal fuzzy number is defined as follows: $A - B = (a_{1} - b_{4}, a_{2} - b_{3}, a_{3} - b_{2}, a_{4} - b_{1}) .$ (10)

Obviously, A - B is still a trapezoidal fuzzy number.

3.2 Prospect theory

As an effective tool to deal with the decision makers’ bounded rationality in decision making process, prospect theory points out that people would make a decision based on the prospect value rather than the expected utility. Unlike expected utility, prospect value depends on reference point and the relative gains or losses but not the absolute outcomes. The prospect value function presented by Tversky and Kahneman [7] is given as follows: $V (Δ x_{i}) = {\begin{matrix} {(Δ x_{i})}^{α}, if Δ x_{i} ⩾ 0, \\ - λ {(- Δ x_{i})}^{β}, if Δ x_{i} < 0 . \end{matrix}$ (11)

Where Δx_i denotes the gain or loss from event a_i relative to the reference point, α and β respectively represent the decision makers’ sensitive degree on gain and loss, λ is the parameter to reflect the decision makers’ risk aversion degree. Generally, we have 0 < α < 1 and 0 < β < 1, which explain the phenomenon called sensitivity decline; In addition, the value of λ is higher than 1, which implies that decision makers are always loss averse. According to Tversky and Kahneman’s empirical research [36], the values of these parameters could be taken as α = β = 0.88, λ = 2.25.

Tversky and Kahneman also stressed that people often distort objective probability. That is, people would overestimate lower probability and underestimate higher probability. Thus, they proposed the probability weighting function to describe this phenomenon: $W (p_{i}) = {\begin{matrix} \frac{p_{i}^{γ}}{{(p_{i}^{γ} + {(1 - p_{i})}^{γ})}^{1 / γ}}, if Δ x_{i} ⩾ 0, \\ \frac{p_{i}^{δ}}{{(p_{i}^{δ} + {(1 - p_{i})}^{δ})}^{1 / δ}}, if Δ x_{i} < 0 . \end{matrix}$ (12)

In the formula, p_i (p_i ⩾ 0) is the objective probability of event a_i, γ and δ represent the decision makers’ attitude coefficients toward the relative gain and loss, respectively. The values of γ and δ denote the distortion degrees of the original probability. The smaller the values are, the severer the distortion. The values given by Tversky and Kahneman [36] are γ = 0.61, δ = 0.69.

Based on the formulas (11) and (12), if there are n possible events a₁, a₂, ⋯ , a_n generated by decision x, and p_i (p_i ⩾ 0) is the objective probability of action a_i, $\sum_{i = 1}^{n} p_{i} = 1$ . Then, the total prospect value of the decision is: $V (x) = \sum_{i = 1}^{n} W (p_{i}) V (Δ x_{i}) .$ (13)

3.3 Fuzzy prospect value function of trapezoidal fuzzy number

In prospect theory, it is important to select a reference point for it could make a great influence on the final decision. Usually, the reference point should be decided at first for the calculation of relative gains or losses. However, a different problem should be solved before deciding the reference point.

The problem is raised based on the premise that the return rates of assets are regarded as generalized trapezoidal fuzzy numbers. Hence, it requires us to find a proper way to transform formula (13) into the form that is suitable for fuzzy numbers. In fact, some researchers have combined prospect theory with fuzzy numbers [5, 37]. Let’s compare their methods and select the better one for portfolio selection.

Method I (for MCDM problem) [5]:

Step 1: Determine the positive ideal solution and the negative ideal solution for each sub-criterion as the reference points.

Step 2: Calculate the relative gain and loss values of each alternative under the criteria by the predetermined distance measure.

Step 3: Determine the positive and negative prospect value matrices.

Step 4: Calculate the corresponding prospect weights.

Step 5: Calculate the integrated prospect values of each alternative and make a ranking.

Method II [37]:

Let R = (r₁, r₂, r₃, r₄) be a trapezoidal fuzzy number, its prospect value function is:

$V (R) = (r_{1}^{α}, r_{2}^{α}, r_{3}^{α}, r_{4}^{α})$ , if r₁ ⩾ 0, r₂ ⩾ 0, r₃ ⩾ 0, r₄ ⩾ 0;

V (R) = (- λ (- r₁) ^β, - λ (- r₂) ^β, - λ (- r₃) ^β, -λ (- r₄) ^β), if r₁ ⩽ 0, r₂ ⩽ 0, r₃ ⩽ 0, r₄ ⩽ 0;

$V (R) = (- λ (- r_{1})^{β}, - λ (- r_{2})^{β}, r_{3}^{α}, r_{4}^{α})$ , if r₁ ⩽ 0, r₂ ⩽ 0, r₃ ⩾ 0, r₄ ⩾ 0;

$V (R) = (- λ (- r_{1})^{β}, r_{2}^{α}, r_{3}^{α}, r_{4}^{α})$ , if r₁ ⩽ 0, r₂ ⩾ 0, r₃ ⩾ 0, r₄ ⩾ 0;

V (R) = (- λ (- r₁) ^β, - λ (- r₂) ^β, - λ (- r₃) ^β, $r_{4}^{α})$ , if r₁ ⩽ 0, r₂ ⩽ 0, r₃ ⩽ 0, r₄ ⩾ 0.

Where parameters α, β and λ are the same as in the formula (11).

In this paper, Method II would be used and extended for portfolio selection with the reasons listed as follows:

It is obvious that the prospect value function in Method II is based on the formula (11), and it is still a trapezoidal fuzzy number that could better reflect the uncertainty of the original.

Method I strongly depends on the predetermined distance measure of fuzzy numbers. Considering that there are many distance measures for trapezoidal fuzzy number [4 , 39], it is difficult for investors to find the appropriate one. Comparatively, the subtraction of trapezoidal fuzzy number is unique and has the rigorous theoretical basis [35]. Therefore, if the reference point (also in the form of trapezoidal fuzzy number) is determined, the relative gain or loss could be obtained uniquely by the subtraction and then the prospect value function could be identified according to Method II.

Suppose the distance measure is predetermined in Method I, and there is only one criterion, the return rate, to select some assets for portfolio. In this case, the weight of the return rate is 1, which means the prospect weight will also be 1 and Step 4 will be meaningless.

Even if investors may choose more than one criterion to select assets for portfolio, it is doubtless that the weight allocation method (such as AHP, BWM and entropy weight method [40 –42]) for each criterion could make a great influence on the result. For investors, it is also a thorny problem to choose an appropriate method to allocate weight for criteria.

In summary, although Method I is effective for MCDM problem, it is unsuitable for portfolio selection problem. Therefore, Method II would be used in this paper. Considering Method II only provides us with the form of prospect value function, in the following sections, we would extend Method II based on the subtraction of trapezoidal fuzzy number and possibility theory.

4 A novel prospect asset filtering frame for portfolio selection

4.1 A novel reference point for comprehensive comparison

As mentioned above, the selection of reference point is important in prospect theory. Many kinds of reference points had been used in some existing studies, such as average, medium point, zero point, optimal point and worst point [8, 43]. In this paper, a novel kind of reference points is put forward.

Firstly, it should be pointed out that the method we use to quantify the return rates of assets as trapezoidal fuzzy numbers is proposed by Vercher and Bermudez [44]. That is, the weekly turns will be transformed into the form of (r₁₀, r₄₀, r₆₀, r₉₀), where r_q is the qth sample percentile of historical returns.

On this basis, if there are n assets A₁, A₂, ⋯ , A_n with the return rates R₁, R₂, ⋯ , R_n in the form of trapezoidal fuzzy number, let S be the set of the historical returns of all the assets, the novel reference point RF is given in the following form: $RF = (S_{10}, S_{40}, S_{60}, S_{90}),$ (14) where S_q is the qth percentile of S.

The feature of the proposed reference point is: if we use the trapezoidal fuzzy number (r_i,10, r_i,40, r_i,60, r_i,90) to quantify the return rate of asset A_i, that is, the possibility distribution of R_i is assumed as a trapezoidal fuzzy number, then, RF can be regarded as the possibility distribution of S, which is equivalent to the joint possibility distribution of R₁, R₂, ⋯ , R_n.

Thus, using RF as a reference point indicates that each asset will be compared with all other assets, which meets investors’ demands and makes the result more comprehensive.

4.2 The fuzzy prospect value of assets

Based on the proposed reference point, the relative gains or losses of each asset can be easily obtained by the subtraction of trapezoidal fuzzy number according to Definition 5.

For the assets A₁, A₂, ⋯ , A_n with the return rates R₁, R₂, ⋯ , R_n in the form of trapezoidal fuzzy number, the relative gain or loss of asset A_i is: $R_{i} - RF = (r_{i, 10} - S_{90}, r_{i, 40} - S_{60}, r_{i, 60} - S_{40}, r_{i, 90} - S_{10}),$ (15) where RF is given by the formula (14).

Then, by Method II, we can get the prospect return rate V (R_i - RF) for asset A_i. According to Definitions 2 and 3, the possibilistic mean value and variance of V (R_i - RF) can also be calculated.

Definition 6. The fuzzy prospect value of asset A_i is defined as: $FV (A_{i}) = \frac{M (V (R_{i} - RF))}{Var (V (R_{i} - RF))} .$ (16)

Remark 1. It should be pointed out that the definition is better than Method I in its meaning for portfolio selection. Considering that possibilistic mean and variance are used as the returns and risks of assets, it is apparent that the formula (16) denotes the prospect return rate of A_i under unit risk in investors’ subjective cognition. Thus, it can better reflect investors’ psychological state.

4.3 The asset filter framework based on prospect theory

It is common that an investor would collect the historical returns of many assets and then choose some assets for investment. In this part, a prospect asset filtering framework is proposed to help investors select assets. For convenience, suppose there are n assets A₁, A₂, ⋯ , A_n with the return rates R₁, R₂, ⋯ , R_n, R_i = (r_1i, r_2i, r_3i, r_4i). If the investor wants to choose m of them to invest in (m ⩽ n), the following operations would be taken:

Exclusion operation: Firstly, let S be the set of the historical returns of all the assets, and then we can obtain the reference point RF = (S₁₀, S₄₀, S₆₀, S₉₀) according to formula (14). Considering an asset with the return rate R = (r₁, r₂, r₃, r₄). When r₃ < S₄₀, it means that the maximum of the interval with membership degree 1 of R is less than the minimum of the interval with membership degree 1 of S. Since membership degree represents investors’ satisfaction degree with the assets, we can exclude the assets which satisfy r_i3 < S₄₀. And then, update the set S and repeat the exclusion operation until all the remaining assets satisfy r_i3 ⩾ S₄₀.

Remark 2. We can ensure that there is at least one asset remaining. In fact, if there is only one asset with the return rate R = (r₁, r₂, r₃, r₄), then we have RF = (S₁₀, S₄₀, S₆₀, S₉₀) = (r₁, r₂, r₃, r₄), which implies r₃ ⩾ S₄₀. In addition, we can ensure that this asset is superior to most of the excluded assets according to the above exclusion operation.

Inclusion operation: For the remaining assets, if there are some assets satisfying r_i2 > S₆₀, similar to the analysis of exclusion operation, it can be considered that these assets are better than others. Therefore, these assets will be picked out and others will be excluded. In this case, the set S would also be updated and both the exclusion operation and the inclusion operation should be repeated until all the remaining assets satisfy r_i2 ⩽ S₆₀.

Remark 3. For any remaining asset A_i with the return rate R = (r_i1, r_i2, r_i3, r_i4), we have r_i1 ⩽ r_i2 ⩽ r_i3 ⩽ r_i4, S₁₀ ⩽ S₄₀ ⩽ S₆₀ ⩽ S₉₀ and r_i2 ⩽ S₆₀, r_i3 ⩾ S₄₀. Thus, by formula (15), the relative gain or loss $R_{i} - RF = (r_{i 1}^{'}, r_{i 2}^{'}, r_{i 3}^{'}, r_{i 4}^{'})$ of asset A_i satisfies $r_{i 1}^{'} ⩽ 0, r_{i 2}^{'} ⩽ 0, r_{i 3}^{'} ⩾ 0, r_{i 4}^{'} ⩾ 0$ , and the corresponding prospect return rate of A_i has the following unified form: $V (R_{i} - RF) = (- λ (- r_{' i 1})^{β}, - λ (- r_{' i 2})^{β}, r_{' i 3}^{α}, r_{' i 4}^{α}),$ (17) where the parameters α, β and λ are the same as in formula (11).

Selection operation: After the inclusion operation, if the number of remaining assets is less than m + 1, the investor would be required to reduce the number of assets to invest in or added new assets. If the investor wants to add other assets, the exclusion operation and the inclusion operation should be taken again. Otherwise, update m. Then, we can ensure that the number of remaining assets is more than m. To choose the assets for investment, the relative fuzzy prospect value of each asset will be calculated by the following formula: $RFV (A_{i}) = FV (A_{i}) - {FV}_{min},$ (18) where FV (A_i) is the fuzzy prospect value of asset A_i and FV_min is the minimum of these fuzzy prospect values. It is apparent that RFV (A_i) represents the prospect return rate of A_i under unit risk relative to the worst remaining asset in investors’ subjective cognition and RFV (A_i) ⩾ 0. Thus, the investor can rank the remaining assets by their relative fuzzy prospect values and then choose the top m to invest in. Usually, the relative fuzzy prospect values are different because it is almost impossible for an asset’s historical data to be exactly the same as others. That is, except for the minimum, the other fuzzy prospect values will be positive.

Remark 4. Formula (18) converts the fuzzy prospect values into non-negative numbers, which not only makes investors feel more intuitive and positive but also is necessary for the next sections.

The prospect asset filtering framework can be briefly summarized in the following steps:

Step 1: Let S be the set of the historical returns of all assets, exclude the assets with r_i3 < S₄₀.

Step 2: Update S, and then repeat Step 1 until all the assets satisfy r_i3 ⩾ S₄₀.

Step 3: Update S, pick out the assets with r_i2 > S₆₀ and exclude all the others.

Step 4: Update S, and then repeat Steps 1 and 2 until all the assets satisfy r_i2 ⩽ S₆₀.

Step 5: If the number of remaining assets is less than m + 1, reduce the number of assets to invest in or add other new assets. If the latter is taken, repeat Steps 1 to 5. Otherwise, update m.

Step 6: Calculate the relative fuzzy prospect values of the remaining assets and then choose the top m to invest in.

Figure 1 shows the process of asset filtering.

Fig. 1

The process of asset filtering.

5 Portfolio selection with the prospect consistency constraint

5.1 The prospect inconsistency coefficient and prospect consistency coefficient

Since we cannot ensure that a selected asset will make profit, it is necessary to invest in multiple assets at the same time to disperse risk. However, how to allocate the investment ratio to each asset is also a problem. As mentioned in introduction, many researchers have studied the problem, but most of them assumed that investors are completely rational and strongly depend on the expected utility theory. In fact, investors would be affected by the historical returns of assets and the comparison between assets. On the one hand, they would have an impression of each asset when they collect the historical data, which would have a great influence on their final decision. On the other hand, they want to make profit and reduce risk as far as possible. For this phenomenon, a new definition called prospect inconsistency coefficient is proposed.

Firstly, in Section 4, the relative fuzzy prospect values of each remaining asset are obtained. These values could be viewed as the scores of each asset in investors’ subjective cognition, which are the quantification of the impressions of each asset. In general, investors would like to allocate more investment to the assets with higher scores. For instance, if there are two assets A and B with the return rates shown in Table 1, investors will allocate more investment ratio to A rather than to B. Especially, investors would like to allocate 75% investment ratio to A and 25% investment ratio to B because A gains three times as many scores as B in investors’ subjective cognition.

Table 1
The return rates of assets A and B

Asset Return 1 Probability 1 Return 2 Probability 2

A 30 0.5 0 0.5

B 10 0.5 0 0.5

Asset	Return 1	Probability 1	Return 2	Probability 2
A	30	0.5	0	0.5
B	10	0.5	0	0.5

It is common that people often distribute their property simply according to the value ratio of things. For example, with the same price, if a concert can bring audiences two times the joy of a football match, people may go to concerts twice a week, but only watch football match once a week. Similarly, we can use the ratio of relative fuzzy prospect value to describe the investors’ preference for assets.

Suppose the relative fuzzy prospect values of the top m selected assets A₁, A₂, ⋯ , A_m are RFV₁, RFV₂, ⋯ , RFV_m, RFV_i ⩾ 0, i = 1, 2, ⋯ , m. Considering that it is almost impossible for an asset’s historical data to be exactly the same as others, and the number of remaining assets is more than m, we can assume that RFV_i > 0, i = 1, 2, ⋯ , m (Otherwise, exclude the assets with RFV_i = 0 and then repeat Step 5 of the prospect asset filtering frame until the condition is satisfied). Then, the prospect investment ratio $x_{i}^{'}$ that investors would allocate to A_i should satisfy that: $\frac{x_{i}^{'}}{x_{j}^{'}} = \frac{{RFV}_{i}}{{RFV}_{j}}, j = 1, 2, \dots, m .$ (19)

It is easy to prove that the formula can be satisfied at the same time for all $x_{i}^{'}, i = 1, 2, \dots, m$ . In fact, if the formula is satisfied when i = i₀, then for any k = 1, 2, ⋯ , m, we have $\frac{x_{k}^{'}}{x_{j}^{'}} = \frac{x_{k}^{'} / x_{i}^{'}}{x_{j}^{'} / x_{i}^{'}} =$ $\frac{{RFV}_{k} / {RFV}_{i}}{{RFV}_{j} / {RFV}_{i}} = \frac{{RFV}_{k}}{{RFV}_{j}}, j = 1, 2, \dots, m$ .

However, investors do not simply allocate investment ratios based on the scores of each asset. They will choose a good portfolio strategy to make great profit while reducing risk as much as possible, if the actual investment ratios do not deviate too much from the prospect investment ratios (or they would not feel well and dismiss the portfolio strategy). The prospect inconsistency coefficient is defined to quantify the deviation between the actual investment ratios and the prospect investment ratios.

Definition 7. Suppose there are m assets A₁, A₂, ⋯ , A_m with the relative fuzzy prospect values RFV₁, RFV₂, ⋯ , RFV_m, RFV_i > 0, i = 1, 2, ⋯ , m. The prospect inconsistency coefficient of portfolio strategy $X = {[x_{1}, x_{2}, \dots, x_{m}]}^{T} (\sum_{i = 1}^{m} x_{i} = 1, x_{i} ⩾ 0)$ is defined as

$\begin{matrix} PIC (X) = max_{i = 1, 2, \dots, m} {PIC}_{i} (X), where {PIC}_{i} (X) \\ = \sqrt{\sum_{j = 1}^{m} {(\frac{x_{j}}{x_{i}} - \frac{{RFV}_{j}}{{RFV}_{i}})}^{2}} . \end{matrix}$ (20)

Remark 5. It is stipulated that $\frac{x_{i}}{x_{j}} = 1 if x_{i} = x_{j} = 0$ for the definition, which is meaningful in reality because the actual investment ratios allocated to assets A_i and A_j are the same. If the relative fuzzy prospect values of A_i and A_j are also the same, the actual investment ratios will meet investors’ preference. Correspondingly, there should be $\frac{x_{j}}{x_{i}} - \frac{{RFV}_{j}}{{RFV}_{i}} = 0$ .

Theorem 1. PIC (X)< + ∞ if and only if X ∈ D, where $D = {^{T} | \sum_{i = 1}^{m} x_{i} = 1, x_{i} > 0}$ .

Proof. Obviously, for any X ∈ D, we have $\frac{x_{j}}{x_{i}} < + \infty$ , i, j = 1, 2, ⋯ , m, and then, PIC_i (X)< + ∞, $PIC (X) = max_{i = 1, 2, \dots, m} {PIC}_{i} (X) < + \infty$ .

It is easy to find that PIC_i (X) ⩾0 (i = 1, 2, ⋯ , m), thus, PIC (X) ⩾ 0.

For any number 0⩽ A < + ∞, if PIC (X) = A, then we have $max_{i = 1, 2, \dots, m} {PIC}_{i} (X) = A$ and PIC_i (X) ⩽ A (i = 1, 2, ⋯ , m). That is, $\sum_{j = 1}^{m} {(\frac{x_{j}}{x_{i}} - \frac{{RFV}_{j}}{{RFV}_{i}})}^{2} ⩽ A^{2}$ , ${(\frac{x_{j}}{x_{i}} - \frac{{RFV}_{j}}{{RFV}_{i}})}^{2} ⩽ A^{2}$ and $| \frac{x_{j}}{x_{i}} - \frac{{RFV}_{j}}{{RFV}_{i}} | ⩽ A (j = 1, 2, \dots, m)$ . According to absolute value inequality, $| \frac{x_{j}}{x_{i}} | ⩽ A + \frac{{RFV}_{j}}{{RFV}_{i}}$ . Since x_i, x_j ⩾ 0 and it is required that $\sum_{i = 1}^{m} x_{i} = 1$ in the definition, we have $| \frac{x_{j}}{x_{i}} | = \frac{x_{j}}{x_{i}}$ and $\frac{1}{x_{i}} = \frac{\sum_{j = 1}^{m} x_{j}}{x_{i}} = \sum_{j = 1}^{m} \frac{x_{j}}{x_{i}} ⩽ \sum_{j = 1}^{m} (A + \frac{{RFV}_{j}}{{RFV}_{i}}),$ (21) so that $x_{i} ⩾ \frac{1}{\sum_{j = 1}^{m} (A + \frac{{RFV}_{j}}{{RFV}_{i}})} > 0 (i = 1, 2, \dots, m)$ , which means X ∈ D.

Remark 6. Based on Theorem 1, if there are any i that satisfies x_i = 0, we have PIC (X) > A, ∀ A > 0.

Theorem 2. The range of PIC (X) is [0, + ∞).

Proof. Let $x_{i} = \frac{{RFV}_{i}}{\sum_{k = 1}^{m} {RFV}_{k}}$ , then x_i > 0 and $\sum_{i = 1}^{m} x_{i} = 1$ . In addition, $\frac{x_{j}}{x_{i}} = \frac{{RFV}_{j}}{{RFV}_{i}}$ , that is, in this case, we have

$\begin{matrix} {PIC}_{i} (X) & = \sqrt{\sum_{j = 1}^{m} {(\frac{x_{j}}{x_{i}} - \frac{{RFV}_{j}}{{RFV}_{i}})}^{2}} \\ = \sqrt{\sum_{j = 1}^{m} {(\frac{{RFV}_{j}}{{RFV}_{i}} - \frac{{RFV}_{j}}{{RFV}_{i}})}^{2}} \\ = 0 (i = 1, 2, \dots, m), \end{matrix}$ (22) and then PIC (X) = 0.

For any number A > 0, let $x_{1} = \frac{1}{B}$ , and $x_{i} = (\frac{A}{\sqrt{m - 1}} + \frac{{RFV}_{i}}{{RFV}_{1}}) \frac{1}{B}, i = 2, 3, \dots, m$ , where $B = \sqrt{m - 1} A + \sum_{j = 1}^{m} \frac{{RFV}_{j}}{{RFV}_{1}}$ . Then, we have B > 0, x_i > 0, and $\begin{matrix} \sum_{i = 1}^{m} x_{i} = \frac{1}{B} + \sum_{i = 2}^{m} (\frac{A}{\sqrt{m - 1}} + \frac{{RFV}_{i}}{{RFV}_{1}}) \frac{1}{B} \\ = \frac{1}{B} [1 + \sum_{i = 2}^{m} (\frac{A}{\sqrt{m - 1}} + \frac{{RFV}_{i}}{{RFV}_{1}})] \\ = \frac{1}{B} (\frac{{RFV}_{1}}{{RFV}_{1}} + \sum_{i = 2}^{m} \frac{A}{\sqrt{m - 1}} + \sum_{i = 2}^{m} \frac{{RFV}_{i}}{{RFV}_{1}}) \\ = \frac{1}{B} (\sqrt{m - 1} A + \sum_{i = 1}^{m} \frac{{RFV}_{i}}{{RFV}_{1}}) = 1 . \end{matrix}$ (23)

In addition, $\begin{matrix} {PIC}_{1} (X) = \sqrt{\sum_{j = 1}^{m} {(\frac{x_{j}}{x_{1}} - \frac{{RFV}_{j}}{{RFV}_{1}})}^{2}} \\ = \sqrt{\sum_{j = 2}^{m} {[(\frac{A}{\sqrt{m - 1}} + \frac{{RFV}_{j}}{{RFV}_{1}}) - \frac{{RFV}_{j}}{{RFV}_{1}}]}^{2}} = A . \end{matrix}$ (24)

If ${PIC}_{1} (X) = max_{i = 1, 2, \dots, m} {PIC}_{i} (X)$ , then we have PIC (X) = PIC₁ (X) = A. Otherwise, $\exists k \neq 1, {PIC}_{k} (X) = max_{i = 1, 2, \dots, m} {PIC}_{i} (X)$ . Suppose that PIC_k (X) = A′, we have PIC (X) = A′.

Since $X \in D = {{[x_{1}, x_{2}, \dots, x_{m}]}^{T} | \sum_{i = 1}^{m} x_{i} = 1, x_{i} > 0}$ and PIC_k is continuous on D, $PIC = max_{i = 1, 2, \dots, m} {PIC}_{i}$ , thus, PIC is also continuous on D with the minimum 0 (when $\frac{x_{j}}{x_{i}} = \frac{{RFV}_{j}}{{RFV}_{i}}$ ). And then, for any A that satisfies 0 < A < A′, ∃X₀ ∈ D such that PIC (X₀) = A.

In conclusion, the range of PIC (X) is [0, + ∞).

Theorem 3. Let X_i = [x_i1, x_i2, ⋯ , x_im] ^T (i = 1, 2, ⋯ , m), where $x_{ij} = {\begin{matrix} 0, j \neq i \\ 1, j = i \end{matrix}$ , then we have PIC_i (X_i) > PIC_j (X_j) if and only if RFV_i < RFV_j.

Proof. For any i = 1, 2, ⋯ , m, we have ${PIC}_{i} (X_{i}) = \sqrt{\sum_{k = 1}^{m} {(\frac{x_{ik}}{x_{ii}} - \frac{{RFV}_{k}}{{RFV}_{i}})}^{2}} = \sqrt{\sum_{k \neq i}^{m} {(\frac{{RFV}_{k}}{{RFV}_{i}})}^{2}}$ , and then ${PIC}_{i} (X_{i}) > {PIC}_{j} (X_{j}) \Leftrightarrow {PIC}_{i}^{2} (X_{i}) > {PIC}_{i}^{2} (X_{j}) \Leftrightarrow \sum_{k \neq i}^{m} {(\frac{{RFV}_{k}}{{RFV}_{i}})}^{2} > \sum_{k \neq j}^{m} {(\frac{{RFV}_{k}}{{RFV}_{j}})}^{2}$ $\Leftrightarrow \sum_{k \neq i}^{m} {({RFV}_{k} {RFV}_{j})}^{2} > \sum_{k \neq j}^{m} {({RFV}_{k} {RFV}_{i})}^{2} \Leftrightarrow \sum_{k = 1}^{m} {({RFV}_{k} {RFV}_{j})}^{2} > \sum_{k = 1}^{m} {({RFV}_{k} {RFV}_{i})}^{2}$ $\Leftrightarrow \sum_{k = 1}^{m} {({RFV}_{k} {RFV}_{j})}^{2} - \sum_{k = 1}^{m} {({RFV}_{k} {RFV}_{i})}^{2} > 0 \Leftrightarrow \sum_{k = 1}^{m} {RFV}_{k}^{2} ({RFV}_{j}^{2} - {RFV}_{i}^{2}) > 0$ $\Leftrightarrow \exists k_{0}, {RFV}_{k 0}^{2} ({RFV}_{j}^{2} - {RFV}_{i}^{2}) > 0 \Leftrightarrow {RFV}_{j}^{2} > {RFV}_{i}^{2} \Leftrightarrow {RFV}_{i} < {RFV}_{j}$ .

Remark 7. We can obtain that PIC_i (X_j) = PIC_j (X_i) > A, ∀ A > 0 for any i ≠ j based on Theorem 1.

Now let us turn back to the definition of prospect inconsistency coefficient. Considering the case that investors could only select one asset to invest in. If the prospect inconsistency coefficient is defined as $PIC (X) = min_{i = 1, 2, \dots, m} {PIC}_{i} (X)$ , we can find that PIC (X_i) = PIC_i (X_i). From Theorem 3, it is obvious that investing in the asset with the maximum relative fuzzy prospect value can bring investors the minimum prospect inconsistency coefficient. In other words, investing in the asset with maximum relative fuzzy prospect value best meets investors’ preference, which is reasonable in reality. Nevertheless, investing in one asset alone cannot disperse risk. From this perspective, no matter which asset is invested in, portfolio strategy X_i cannot provide investors with enough sense of security. That is, we should have PIC (X_i) = PIC (X_j) =+ ∞ for any i ≠ j, which is the reason that we defined $PIC (X) = max_{i = 1, 2, \dots, m} {PIC}_{i} (X)$ .

Remark 8. In order to simplify the model proposed in the next part, the definition of $PIC (X) = \frac{1}{m} \sum_{i = 1}^{m} {PIC}_{i} (X)$ is not taken into account in this paper.

Since the range of PIC (X) is [0, + ∞), which cannot provide investors with an intuitive appreciation, the definition of prospect consistency coefficient is presented as follows.

Definition 7. Suppose there are m assets A₁, A₂, ⋯ , A_m with the relative fuzzy prospect values RFV₁, RFV₂, ⋯ , RFV_m, RFV_i > 0, i = 1, 2, ⋯ , m. The prospect consistency coefficient of portfolio strategy $X = {[x_{1}, x_{2}, \dots, x_{m}]}^{T} (\sum_{i = 1}^{n} x_{i} = 1, x_{i} ⩾ 0)$ is defined in the following: $PCC (X) = \frac{1}{1 + PIC (X)},$ (25) where PIC (X) denotes the corresponding prospect inconsistency coefficient.

The relationship between PIC (X) and PCC (X) is similar to that between similarity measure and distance measure [4].

According to Theorems 1 to 3, we have

Theorem 4. PCC (X) =0 if and only if X ∈ D, where $D = {{[x_{1}, x_{2}, \dots, x_{m}]}^{T} | \sum_{i = 1}^{m} x_{i} = 1, \exists i, x_{i} = 0}$ .

Theorem 5. PCC (X) = 1 if and only if $X = \frac{1}{\sum_{k = 1}^{m} {RFV}_{k}} {[{RFV}_{1}, {RFV}_{2}, \dots, {RFV}_{m}]}^{T}$ .

Theorem 6. The range of PCC (X) is [0, 1].

Prospect consistency coefficient can describe the consistency between a portfolio strategy and investors’ preference with range [0, 1]. In the next part, we will construct a portfolio selection model with prospect consistency constraint to provide investors with a satisfactory portfolio strategy while making profit and reducing risk as much as possible.

5.2 The possibilistic portfolio model with prospect consistency constraint

Based on possibility theory, we can get the expected return and risk of a portfolio strategy by possibilistic mean value and variance. Many researchers constructed models to maximize possibilistic mean value while minimizing possibilistic variance to get an optimal portfolio strategy. However, as mentioned above, if a portfolio strategy strongly deviates from investors’ preference, it would not be adopted because it conflicts with investors’ intuition and cannot gain their trust. For this reason, a possibilistic portfolio model with prospect consistency constraint is proposed. The model takes the prospect consistency coefficient as constraint to reflect investors’ tolerance level of the deviation between the portfolio strategy and their preference. And then, within investors’ tolerance level, the model would provide them with an optimal strategy.

The model is constructed as follows. $\begin{matrix} max M (\sum_{i = 1}^{n} x_{i} R_{i}) \\ min Var (\sum_{i = 1}^{n} x_{i} R_{i}) \end{matrix}$ $\begin{matrix} s . t . {\begin{matrix} \sum_{i = 1}^{n} x_{i} = 1, \\ PCC (X) ⩾ δ, \\ x_{i} ⩾ 0, \\ i = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (26)

Where X = [x₁, x₂, ⋯ , x_n] ^T is the portfolio strategy, x_i is the investment ratio of A_i, R_i = (r_1i, r_2i, r_3i, r_4i) is the return rate of A_i in trapezoidal fuzzy number form, $M (\sum_{i = 1}^{n} x_{i} R_{i})$ is the possibilistic mean value of the strategy and denotes return, $Var (\sum_{i = 1}^{n} x_{i} R_{i})$ is the possibilistic variance of the strategy and denotes risk, PCC (X) is the prospect consistency coefficient of the strategy, δ is the parameter which represents investors’ tolerance level.

According to Theorem 6, the range of parameter δ is [0, 1]. It is apparent that the greater the value of δ, the lower the investors’ tolerance level. When δ = 0, the prospect consistency constraint is meaningless, and Model (26) regresses to the general possibilistic M-V model.

By Properties 1 and 2, the formulas (3) and (7), and the definition of PCC (X), Model (26) can be rewritten as $\begin{matrix} max \frac{1}{6} \sum_{i = 1}^{n} x_{i} [2 (r_{i 2} + r_{i 3}) + (r_{i 1} + r_{i 4})] \\ min \frac{1}{24} \sum_{i, j = 1}^{n} x_{i} x_{j} [2 (r_{i 3} - r_{i 2}) (r_{j 3} - r_{j 2}) \\ + (r_{i 4} + r_{i 3} - r_{i 2} - r_{i 1}) (r_{j 4} + r_{j 3} - r_{j 2} - r_{j 1})] \\ s . t . {\begin{matrix} \sum_{i = 1}^{n} x_{i} = 1, \\ \frac{1}{1 + max_{i = 1, 2, \dots, n} {PIC}_{i} (X)} ⩾ δ, \\ {PIC}_{i} (X) = \sqrt{\sum_{j = 1}^{n} {(\frac{x_{j}}{x_{i}} - \frac{{RFV}_{j}}{{RFV}_{i}})}^{2}}, \\ x_{i} ⩾ 0, i = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (27)

Where RFV_i is the relative fuzzy prospect value of A_i.

Considering that there are too many constraints PIC_i (X) of the model, which would make the model over-complicated and inflexible, it is better to simplify the model. Similar to the BWM method and AHP method [40, 41], in reality, investors will not compare all the assets in pairs. They prefer to select the best one and the worst one, and then compare each asset with these two assets. Therefore, denote RFV_best and RFV_worst as the maximum and minimum relative fuzzy prospect values, respectively, and then let x_best, x_worst be the investment ratios allocated to the assets with RFV_best and RFV_worst. Model (27) can be simplified as follows. $\begin{matrix} max \frac{1}{6} \sum_{i = 1}^{n} x_{i} [2 (r_{i 2} + r_{i 3}) + (r_{i 1} + r_{i 4})] \\ min \frac{1}{24} \sum_{i, j = 1}^{n} x_{i} x_{j} [2 (r_{i 3} - r_{i 2}) (r_{j 3} - r_{j 2}) \\ + (r_{i 4} + r_{i 3} - r_{i 2} - r_{i 1}) (r_{j 4} + r_{j 3} - r_{j 2} - r_{j 1})] \\ s . t . {\begin{matrix} \sum_{i = 1}^{n} x_{i} = 1, \\ \frac{1}{1 + max {{PIC}_{best} (X), {PIC}_{worst} (X)}} ⩾ δ, \\ {PIC}_{best} (X) = \sqrt{\sum_{j = 1}^{n} {(\frac{x_{j}}{x_{best}} - \frac{{RFV}_{j}}{{RFV}_{best}})}^{2}}, \\ {PIC}_{worst} (X) = \sqrt{\sum_{j = 1}^{n} {(\frac{x_{j}}{x_{worst}} - \frac{{RFV}_{j}}{{RFV}_{worst}})}^{2}}, \\ x_{worst} ⩽ x_{i} ⩽ x_{best}, i = 1, 2, \dots, n, \\ x_{worst} ⩾ 0 . \end{matrix} \end{matrix}$ (28)

Notice that there is a new constraint x_worst ⩽ x_i ⩽ x_best in Model (28). This constraint satisfies investors’ preference according to formula (19) and will make the portfolio strategy more acceptable.

To solve the proposed model, let $D = {{[x_{1}, x_{2}, \dots, x_{n}]}^{T} | \sum_{i = 1}^{n} x_{i} = 1, x_{i} ⩾ 0 .}$ , M_max and M_min be the maximum and minimum of $M (\sum_{i = 1}^{n} x_{i} R_{i})$ when X ∈ D, Var_max and Var_min be the maximum and minimum of $Var (\sum_{i = 1}^{n} x_{i} R_{i})$ when X ∈ D. Then, the objective functions of Model (28) are equivalent to the following objectives: $max \frac{M (\sum_{i = 1}^{n} x_{i} R_{i}) - M_{min}}{M_{max} - M_{min}},$ (29) $max \frac{{Var}_{max} - Var (\sum_{i = 1}^{n} x_{i} R_{i})}{{Var}_{max} - {Var}_{min}} .$ (30)

In addition, it is easy to find that the range of these new objective functions is [0, 1]. It can be considered that the new objective functions represent the degrees that the original objective functions reached when X ∈ D. The ideal strategy is to make the degrees reach 1. However, it is impossible because of the conflict between risk and return, so that it is necessary to balance the new objective functions. In this paper, the method to balance the objectives is to transform Model (28) into the following single objective model: $\begin{matrix} max θ \\ s . t . {\begin{matrix} min (\frac{M (\sum_{i = 1}^{n} x_{i} R_{i}) - M_{min}}{M_{max} - M_{min}}, \\ \frac{{Var}_{max} - Var (\sum_{i = 1}^{n} x_{i} R_{i})}{{Var}_{max} - {Var}_{min}}) ⩾ θ, \\ M (\sum_{i = 1}^{n} x_{i} R_{i}) = \frac{1}{6} \sum_{i = 1}^{n} x_{i} [2 (r_{i 2} + r_{i 3}) + (r_{i 1} + r_{i 4})], \\ Var (\sum_{i = 1}^{n} x_{i} R_{i}) = \frac{1}{24} \sum_{i, j = 1}^{n} x_{i} x_{j} [2 (r_{i 3} - r_{i 2}) (r_{j 3} - r_{j 2}) \\ + (r_{i 4} + r_{i 3} - r_{i 2} - r_{i 1}) (r_{j 4} + r_{j 3} - r_{j 2} - r_{j 1})], \\ \frac{1}{1 + max {{PIC}_{best} (X), {PIC}_{worst} (X)}} ⩾ δ, \\ {PIC}_{best} (X) = \sqrt{\sum_{j = 1}^{n} {(\frac{x_{j}}{x_{best}} - \frac{{RFV}_{j}}{{RFV}_{best}})}^{2}}, \\ {PIC}_{worst} (X) = \sqrt{\sum_{j = 1}^{n} {(\frac{x_{j}}{x_{worst}} - \frac{{RFV}_{j}}{{RFV}_{worst}})}^{2}}, \\ x_{worst} ⩽ x_{i} ⩽ x_{best}, i = 1, 2, \dots, n, \\ \sum_{i = 1}^{n} x_{i} = 1, x_{worst} ⩾ 0 . \end{matrix} \end{matrix}$ (31)

We can ensure that the feasible region of Model (31) is not empty. In fact, let $X = \frac{1}{\sum_{k = 1}^{n} {RFV}_{k}} {[{RFV}_{1}, {RFV}_{2}, \dots, {RFV}_{n}]}^{T}$ be a portfolio strategy, X is a feasible solution of the model with PCC (X) = 1 and 0 < x_worst ⩽ x_i ⩽ x_best (i = 1, 2, …, n).

For convenience, if we denote $\frac{M (\sum_{i = 1}^{n} x_{i} R_{i}) - M_{min}}{M_{max} - M_{min}}, \frac{{Var}_{max} - Var (\sum_{i = 1}^{n} x_{i} R_{i})}{{Var}_{max} - {Var}_{min}}$ as f₁ (X) , f₂ (X) respectively, the first constraint in Model (31) can be written as min { f₁ (X) , f₂ (X) } ⩾ θ. Suppose that the optimal solution of Model (31) is X₀, θ₀, we have min { f₁ (X₀) , f₂ (X₀) } = θ₀. If there is a Pareto solution X′ better than X₀ and satisfies f₁ (X′) > f₁ (X₀) , f₂ (X′) > f₂ (X₀), let θ′ = min{ f₁ (X′) , f₂ (X′) }, it is trivial to get θ′ > θ₀. In other words, the optimal solution of Model (31) should be X′, θ′ but not X₀, θ₀, which is contrary to our hypothesis; However, it is possible to find a solution X″, θ″ that satisfies f₁ (X₁) > f₁ (X₀) , f₂ (X₁) = f₂ (X₀) or f₁ (X₁) = f₁ (X₀) , f₂ (X₁) > f₂ (X₀), which means that the optimal solution of Model (31) may not be a Pareto solution. In fact, due to the form of prospect consistency constraint, the domain D′ may be non-convex, discrete and very complex. Hence, it is too hard to find the Pareto solutions of Model (31) by the existing methods (Such as the constraint method in [45]).

6 Numerical application

6.1 Asset filtering process

In order to verify the validity of the proposed model, in this section, a numerical example is provided.

Suppose there are 10 assets A₁, A₂, ⋯ , A₁₀ for portfolio selection and an investor wants to choose six of them for investment. To make the example more realistic, the weekly returns of some securities are used, including FTSE 100 Index, DAX 30 Index, CAC 40 Index, AEX Index, BEL 20 Index, SMI Index, PSI 20 Index, S&P 500 Index, KLCI (KLSE) Index and Nikkei 225 Index. All the historical data are from January 4, 2008 to July 26, 2019, and the method to quantify the return rates of assets as trapezoidal fuzzy number is proposed by Vercher and Bermudez [44]. Then, the return rates are listed in Table 2.

Table 2
The return rates of assets A₁, A₂, ⋯ , A₁₀

Return rates (R_i) r _i1 r _i2 r _i3 r _i4

FTSE 100 Index (A₁) –2.5261 –0.2894 0.6568 2.6572

DAX 30 Index (A₂) –3.5217 –0.2251 1.0188 3.4137

CAC 40 Index (A₃) –3.5045 –0.3332 0.8699 3.4074

AEX Index (A₄) –3.1386 –0.2443 0.8239 3.1676

BEL 20 Index (A₅) –3.2027 –0.3956 0.8563 2.9969

SMI Index (A₆) –2.6804 –0.2008 0.7764 2.5120

PSI 20 Index (A₇) –3.5285 –0.4984 0.5419 3.2243

S&P 500 Index (A₈) –2.6185 –0.0819 0.6886 2.6702

KLCI Index (A₉) –1.6694 –0.2166 0.4216 1.7470

Nikkei 225 Index (A₁₀) –3.4489 –0.4320 0.9563 3.5308

Reference point (RF) –3.0376 –0.2910 0.7448 2.9777

Return rates (R_i)	r _i1	r _i2	r _i3	r _i4
FTSE 100 Index (A₁)	–2.5261	–0.2894	0.6568	2.6572
DAX 30 Index (A₂)	–3.5217	–0.2251	1.0188	3.4137
CAC 40 Index (A₃)	–3.5045	–0.3332	0.8699	3.4074
AEX Index (A₄)	–3.1386	–0.2443	0.8239	3.1676
BEL 20 Index (A₅)	–3.2027	–0.3956	0.8563	2.9969
SMI Index (A₆)	–2.6804	–0.2008	0.7764	2.5120
PSI 20 Index (A₇)	–3.5285	–0.4984	0.5419	3.2243
S&P 500 Index (A₈)	–2.6185	–0.0819	0.6886	2.6702
KLCI Index (A₉)	–1.6694	–0.2166	0.4216	1.7470
Nikkei 225 Index (A₁₀)	–3.4489	–0.4320	0.9563	3.5308
Reference point (RF)	–3.0376	–0.2910	0.7448	2.9777

According to Section 4.1, the reference point is also quantified and shown in Table 2. Next, we will use the prospect asset filtering frame to filter these assets.

Exclusion operation: Since r_i3 > 0 > S₄₀ = -0 . 2910 for all assets, no assets should be excluded.

Inclusion operation: Since r_i2 < 0 < S₆₀ = 0 .7448 for all assets, no assets should be excluded either.

Selection operation: Since the number of remaining assets is more than six, we just need to calculate the relative fuzzy prospect values of each asset and then select the top six to invest in.

According to formulas (15) and (17), the prospect return rates of each asset are calculated and listed in Table 3. Here, the values of parameters α, β and λ are taken as α = β = 0.88 and λ = 2.25, which is given by Tversky and Kahneman [36].

Table 3

The prospect return rates of assets A₁, A₂, ⋯ , A₁₀

Prospect return rates ( $R_{i}^{'}$ )	$r_{i 1}^{'}$	$r_{i 2}^{'}$	$r_{i 3}^{'}$	$r_{i 4}^{'}$
FTSE 100 Index (A₁)	–10.0917	–2.3175	0.9539	4.6219
DAX 30 Index (A₂)	–11.6818	–2.1902	1.2680	5.1581
CAC 40 Index (A₃)	–11.6546	–2.4036	1.1403	5.1536
AEX Index (A₄)	–11.0737	–2.2284	1.1004	4.9845
BEL 20 Index (A₅)	–11.1758	–2.5258	1.1285	4.8636
SMI Index (A₆)	–10.3402	–2.1419	1.0591	4.5180
PSI 20 Index (A₇)	–11.6925	–2.7251	0.8514	5.0246
S&P 500 Index (A₈)	–10.2406	–1.9029	0.9820	4.6312
KLCI Index (A₉)	–8.6956	–2.1733	0.7422	3.9652
Nikkei 225 Index (A₁₀)	–11.5665	–2.5965	1.2147	5.2403

According to formulas (3), (5), (16) and (18), the prospect possibilistic mean values $M (R_{i}^{'})$ , prospect possibilistic variances $Var (R_{i}^{'})$ , fuzzy prospect values FV_i and relative fuzzy prospect values RFV_i of each asset are calculated and listed in Table 4.

Table 4

$M (R_{i}^{'})$ , $Var (R_{i}^{'})$ , FV_i, RFV_i and the ranking of assets A₁, A₂, ⋯ , A₁₀

Assets	$M (R_{i}^{'})$	$Var (R_{i}^{'})$	FV _i	RFV _i	Ranking
FTSE 100 Index (A₁)	–1.3662	14.3693	–0.0951	0.0219	8
DAX 30 Index (A₂)	–1.3947	18.1639	–0.0768	0.0402	1
CAC 40 Index (A₃)	–1.5046	18.3053	–0.0822	0.0348	3
AEX Index (A₄)	–1.3908	16.5840	–0.0839	0.0331	4
BEL 20 Index (A₅)	–1.5178	17.2728	–0.0879	0.0291	5
SMI Index (A₆)	–1.3313	14.4429	–0.0922	0.0248	7
PSI 20 Index (A₇)	–1.7359	18.2253	–0.0952	0.0217	9
S&P 500 Index (A₈)	–1.2419	13.8311	–0.0898	0.0272	6
KLCI Index (A₉)	–1.2655	10.8174	–0.1170	0.0000	10
Nikkei 225 Index (A₁₀)	–1.5149	18.9230	–0.0801	0.0369	2

The ranking of assets based on the relative fuzzy prospect values is also shown in Table 4. From Table 4, we know that DAX 30 Index, CAC 40 Index, AEX Index, BEL 20 Index, S&P 500 Index and Nikkei 225 Index will be selected to invest in. In addition, we can find that A₂ is the asset with the maximum relative fuzzy prospect value and A₈ is the selected asset with the minimum relative fuzzy prospect value.

6.2 Portfolio Selection with prospect consistency constraint

Suppose the investor’s tolerance level is 0.2, that is, δ = 0.8. To make profit and reduce risk as much as possible, a portfolio selection model is constructed based on Model (31) as follows. $\begin{matrix} max θ \\ s . t . {\begin{matrix} min (\frac{M (\sum_{i = 1}^{6} x_{i} R_{i}) - 0.1192}{0.2466 - 0.1192}, \frac{3.0782 - Var (\sum_{i = 1}^{6} x_{i} R_{i})}{3.0782 - 1.579}) ⩾ θ, \\ M (\sum_{i = 1}^{6} x_{i} R_{i}) = 0 . 2466 x_{1} + 0 . 1627 x_{2} + 0 . 1980 x_{3} + 0 . 1192 x_{4} + 0 . 2109 x_{5} + 0 . 1884 x_{6}, \\ Var (\sum_{i = 1}^{6} x_{i} R_{i}) = X^{T} CX, \\ \frac{1}{1 + max {{PIC}_{best} (X), {PIC}_{worst} (X)}} ⩾ 0.8, \\ {PIC}_{best} (X) = \sqrt{{(\frac{x_{2}}{x_{1}} - \frac{0 . 0348}{0 . 0402})}^{2} + {(\frac{x_{3}}{x_{1}} - \frac{0 . 0331}{0 . 0402})}^{2} + {(\frac{x_{4}}{x_{1}} - \frac{0 . 0291}{0 . 0402})}^{2} + {(\frac{x_{5}}{x_{1}} - \frac{0 . 0272}{0 . 0402})}^{2} + {(\frac{x_{6}}{x_{1}} - \frac{0 . 0369}{0 . 0402})}^{2}}, \\ {PIC}_{worst} (X) = \sqrt{{(\frac{x_{1}}{x_{5}} - \frac{0 . 0402}{0 . 0272})}^{2} + {(\frac{x_{2}}{x_{5}} - \frac{0 . 0348}{0 . 0272})}^{2} + {(\frac{x_{3}}{x_{5}} - \frac{0 . 0331}{0 . 0272})}^{2} + {(\frac{x_{4}}{x_{5}} - \frac{0 . 0291}{0 . 0272})}^{2} + {(\frac{x_{6}}{x_{5}} - \frac{0 . 0369}{0 . 0272})}^{2}}, \\ x_{1} ⩽ x_{i} ⩽ x_{5}, i = 1, 2, \dots, 6, \\ \sum_{i = 1}^{6} x_{i} = 1, x_{5} ⩾ 0 . \end{matrix} \end{matrix}$ (32)

Where X = [x₁, x₂, x₃, x₄, x₅, x₆] ^T are the investment ratios of assets A₂, A₃, A₄, A₅, A₈, A₁₀, X^T is the transpose of X and C is the covariance matrix of these assets, $C = (\begin{matrix} 2 . 91652 . 89032 . 62402 . 66932 . 14482 . 9957 \\ 2 . 89032 . 86452 . 60062 . 64512 . 12602 . 9686 \\ 2 . 62402 . 60062 . 36102 . 40101 . 93042 . 6948 \\ 2 . 66932 . 64512 . 40102 . 44421 . 96162 . 7429 \\ 2 . 14482 . 12601 . 93041 . 96161 . 57922 . 2017 \\ 2 . 99572 . 96862 . 69482 . 74292 . 20173 . 0782 \end{matrix}) .$ (33)

For comparison, the possibilistic M-V model is also constructed as follows. $\begin{matrix} max M (\sum_{i = 1}^{6} x_{i} R_{i}) = 0 . 2466 x_{1} + 0 . 1627 x_{2} \\ + 0 . 1980 x_{3} + 0 . 1192 x_{4} + 0 . 2109 x_{5} + 0 . 1884 x_{6} \\ min Var (\sum_{i = 1}^{6} x_{i} R_{i}) = X^{T} CX \\ s . t . {\begin{matrix} \sum_{i = 1}^{6} x_{i} = 1, \\ x_{i} ⩾ 0, i = 1, 2, \dots, 6 . \end{matrix} \end{matrix}$ (34)

Model (34) will be transformed into a single objective model in the same way as Model (28). And then, we input the data into Lingo software to obtain the optimal solutions of these two models, which are shown in Table 5.

Table 5

The optimal solutions of Models (32) and (34)

Assets	A ₂	A ₃	A ₄	A ₅	A ₈	A ₁₀
The proposed model	0.1807	0.1558	0.1408	0.1667	0.1894	2.5467
The general model	0	0	0.7387	0	0.2202	0

Based on Table 5, the returns (mean), risks (variance), Shannon entropy [34], the values of the objective θ and the prospect consistency coefficient $PCC = \frac{1}{1 + max {{PIC}_{best} (X), {PIC}_{worst} (X)}}$ are calculated and shown in Table 6.

Table 6

The returns, risks, Shannon entropy, θ and PCC of Models (32) and (34)

Assets	Return	Risk	Shannon entropy	θ	PCC
The proposed model	0.1894	2.5467	1.7870	0.3544	0.8000
The general model	0.2202	1.8888	0.5744	0.7933	0.2685

The concept of entropy was originally derived from thermo-dynamics to describe the disorder of a system. Later, for its ability to measure diversity, it is widely used in many fields including portfolio selection. The formula of Shannon entropy is given as follows. $E (X) = - \sum_{i = 1}^{6} x_{i} ln x_{i}$ (35)

The greater the entropy is, the higher the diversity. The higher the diversity is, the greater the risk dispersion.

From Table 6, we can find that:

The return and risk of the optimal strategy obtained by our proposed model are worse than that of the general model. Similarly, the value of objective θ, the degree that the original objective functions ( $max M (\sum_{i = 1}^{6} x_{i} R_{i}), min Var (\sum_{i = 1}^{6} x_{i} R_{i})$ ) reached, is far less than that of the general model. The reason is that the constraint of our proposed model is more than that of the general. Although the return and risk of the proposed model are worse, it should be noticed that the value of PCC of our strategy is far higher than that of the general model. That is, the strategy provided by our model can better meet the investor’s preference.

The Shannon entropy of the proposed model is far higher than that of the general. In fact, from Table 5, we can see that the portfolio strategy obtained by our model is more diverse. Comparatively, the portfolio strategy obtained by the general model only allocates the investment ratios to two assets, which shows the poor ability to disperse risk.

Although A₂ is the asset with the maximum relative fuzzy prospect value, from Table 5, we can see that the general model does not allocate any investment ratio to A₂, which is inconsistent with the investor’s preference and then leads to a small value of PCC.

In summary, compared with the general model, our model can provide investors with a more satisfactory portfolio strategy and better disperse risk.

7 Sensitivity analysis and comparison

In this section, the sensitivity of parameters α, β and λ in formula (11) and parameter δ in Model (31) will be analyzed to study their properties and then help investors choose appropriate values according to their preferences. For convenience, the data used in Section 6 will also be used in the following.

7.1 The sensitivity analysis of parameters α, β and λ

Considering the values of α and β are the same (0.88) based on Tversky and Kahneman’s empirical research [36], in this part, we will keep α = β in all times. And then, changing the value of α from 0.1 to 0.9 while changing the value of λ from 1.1 to 2.9. The results are shown in Tables 7 9 and Figs. 3 5.

Table 7
The returns of Model (31) with δ = 0.8 and α = β

Returns α = 0.1 α = 0.2 α = 0.3 α = 0.4 α = 0.5 α = 0.6 α = 0.7 α = 0.8 α = 0.9

λ = 1.1 0.1977 0.1977 0.1977 0.1976 0.1976 0.1977 0.1995 0.1996 0.1997

λ = 1.3 0.1985 0.1986 0.1987 0.1988 0.1989 0.1989 0.1989 0.1988 0.1988

λ = 1.5 0.1988 0.1988 0.1987 0.1987 0.1986 0.1985 0.1985 0.1984 0.1925

λ = 1.7 0.1985 0.1985 0.1984 0.1923 0.1922 0.1921 0.1920 0.1919 0.1915

λ = 1.9 0.1838 0.1907 0.1906 0.1906 0.1916 0.1915 0.1914 0.1913 0.1912

λ = 2.1 0.1834 0.1834 0.1901 0.1900 0.1900 0.1899 0.1898 0.1898 0.1896

λ = 2.3 0.1831 0.1831 0.1831 0.1896 0.1896 0.1895 0.1895 0.1894 0.1893

λ = 2.5 0.1828 0.1829 0.1829 0.1829 0.1892 0.1892 0.1891 0.1891 0.1890

λ = 2.7 0.1826 0.1826 0.1827 0.1827 0.1827 0.1889 0.1888 0.1888 0.1887

λ = 2.9 0.1823 0.1824 0.1824 0.1825 0.1825 0.1825 0.1886 0.1886 0.1563

Returns	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9
λ = 1.1	0.1977	0.1977	0.1977	0.1976	0.1976	0.1977	0.1995	0.1996	0.1997
λ = 1.3	0.1985	0.1986	0.1987	0.1988	0.1989	0.1989	0.1989	0.1988	0.1988
λ = 1.5	0.1988	0.1988	0.1987	0.1987	0.1986	0.1985	0.1985	0.1984	0.1925
λ = 1.7	0.1985	0.1985	0.1984	0.1923	0.1922	0.1921	0.1920	0.1919	0.1915
λ = 1.9	0.1838	0.1907	0.1906	0.1906	0.1916	0.1915	0.1914	0.1913	0.1912
λ = 2.1	0.1834	0.1834	0.1901	0.1900	0.1900	0.1899	0.1898	0.1898	0.1896
λ = 2.3	0.1831	0.1831	0.1831	0.1896	0.1896	0.1895	0.1895	0.1894	0.1893
λ = 2.5	0.1828	0.1829	0.1829	0.1829	0.1892	0.1892	0.1891	0.1891	0.1890
λ = 2.7	0.1826	0.1826	0.1827	0.1827	0.1827	0.1889	0.1888	0.1888	0.1887
λ = 2.9	0.1823	0.1824	0.1824	0.1825	0.1825	0.1825	0.1886	0.1886	0.1563

Table 8

The risks of Model (31) with δ = 0.8 and α = β

Risks	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9
λ = 1.1	2.3324	2.3317	2.3312	2.3310	2.3312	2.3317	2.3471	2.3465	2.3462
λ = 1.3	2.3587	2.3588	2.3591	2.3594	2.3608	2.3631	2.3661	2.3698	2.3743
λ = 1.5	2.3748	2.3753	2.3763	2.3777	2.3795	2.3819	2.3849	2.3885	2.5040
λ = 1.7	2.3894	2.3899	2.3907	2.5109	2.5105	2.5104	2.5105	2.5110	2.5556
λ = 1.9	2.5572	2.5410	2.5396	2.5383	2.5174	2.5170	2.5168	2.5169	2.5173
λ = 2.1	2.5617	2.5629	2.5484	2.5467	2.5452	2.5440	2.5432	2.5428	2.5428
λ = 2.3	2.5657	2.5667	2.5678	2.5539	2.5519	2.5503	2.5490	2.5482	2.5478
λ = 2.5	2.5692	2.5700	2.5710	2.5721	2.5577	2.5556	2.5540	2.5527	2.5520
λ = 2.7	2.5724	2.5731	2.5738	2.5747	2.5756	2.5603	2.5582	2.5566	2.5555
λ = 2.9	2.5752	2.5758	2.5764	2.5770	2.5777	2.5785	2.5619	2.5599	2.7108

Table 9

The Shannon entropy of Model (31) with δ = 0.8 and α = β

Shannon entropy	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9
λ = 1.1	1.7836	1.7836	1.7837	1.7839	1.7840	1.7840	1.7768	1.7762	1.7758
λ = 1.3	1.7836	1.7830	1.7825	1.7822	1.7820	1.7820	1.7823	1.7825	1.7826
λ = 1.5	1.7832	1.7833	1.7836	1.7838	1.7841	1.7843	1.7844	1.7843	1.7840
λ = 1.7	1.7845	1.7847	1.7848	1.7846	1.7850	1.7853	1.7856	1.7860	1.7834
λ = 1.9	1.7846	1.7848	1.7852	1.7856	1.7865	1.7868	1.7870	1.7873	1.7876
λ = 2.1	1.7853	1.7852	1.7852	1.7857	1.7861	1.7864	1.7867	1.7870	1.7872
λ = 2.3	1.7856	1.7855	1.7854	1.7853	1.7858	1.7862	1.7865	1.7868	1.7870
λ = 2.5	1.7857	1.7856	1.7854	1.7853	1.7853	1.7857	1.7861	1.7865	1.7867
λ = 2.7	1.7857	1.7855	1.7854	1.7852	1.7850	1.7852	1.7857	1.7861	1.7863
λ = 2.9	1.7855	1.7853	1.7852	1.7850	1.7848	1.7846	1.7852	1.7856	1.7852

Fig. 2

The asset selection distribution.

Fig. 3

The returns of Model (31) with δ = 0.8 and α = β.

Fig. 4

The risks of Model (31) with δ = 0.8 and α = β.

Fig. 5

The Shannon entropy of Model (31) with δ = 0.8 and α = β.

It should be noticed that when the values of α, β and λ are changed, the prospect return rates of each asset are also changed. In other words, the prospect possibilistic mean values, prospect possibilistic variances, fuzzy prospect values and relative fuzzy prospect values are all changed so that the ranking of assets is changed and the assets selected to invest in would be different. The asset selection distribution is shown in Fig. 2.

From Fig. 2, we can find that:

In all the cases, the investment ratios of assets A₁, A₉ are all zero, which indicates that neither of these two assets has ever been selected to invest in. A₇ is similar to them. In fact, A₇ is selected only once when α, β = 0.9 and λ = 2.9, which is the reason that return decreases sharply and risk increases sharply in that case (see Figs. 3 and 4).

In all the cases, the investment ratios of assets A₂, A₃, A₄, A₈, A₁₀ are higher than 0.1, which indicates that these assets have a good performance and are not affected by the parameters. In addition, we can find that the investment ratio of A₂ is the highest at all times. According to the constraint x_worst ⩽ x_i ⩽ x_best (i = 1, 2, ⋯ , n) of Model (31), it can be inferred that A₂ is the asset with the maximum relative fuzzy prospect value no matter how the parameters change.

Based on (i) and (ii), it is not difficult to find that either A₅ or A₆ is the last selected asset in most cases. The minimum investment ratio allocated to these two assets is zero, which means that sometimes they are not selected. In fact, when A₅ is selected, A₆ is excluded; when A₆ is selected, A₅ is excluded.

From Tables 7 9 and Figs. 3 5, we can find that:

In most cases, the return obtained by the proposed method increases if λ decreases. Comparatively, the sensitivity of return to parameters α and β is far less than that to λ. In addition, when 0.1 ⩽ α, β ⩽ 0.3, there is a sharp decrease of return from λ = 1.7 to λ = 1.9; when 0.3 ⩽ α, β ⩽ 0.8, there is a sharp decrease of return from λ = 1.5 to λ = 1.7; when α, β = 0.9, there is a sharp decrease of return from λ = 1.3 to λ = 1.5. That is, the sharp decreasing point λ₀ decreases when α and β increase.

In most cases, the risk obtained by the proposed method increases if λ increases. Comparatively, the sensitivity of risk to parameters α and β is far less than that to λ. In addition, when 0.1 ⩽ α, β ⩽ 0.3, there is a sharp increase of risk from λ = 1.7 to λ = 1.9; when 0.3 ⩽ α, β ⩽ 0.8, there is a sharp increase of risk from λ = 1.5 to λ = 1.7; when α, β = 0.9, there is a sharp increase of risk from λ = 1.3 to λ = 1.5. That is, the sharp increasing point λ₀ decreases when α and β increase, which is similar to (i).

In most cases, when λ ⩾ 1.9 and $α, β ⩾ \frac{λ - 1.7}{2}$ , the Shannon entropy of the portfolio strategy obtained by the proposed method increases if α and β increase; when λ ⩾ 1.9 and $α, β ⩽ \frac{λ - 1.7}{2}$ , the Shannon entropy decreases if α and β increase. In addition, when 0.1 ⩽ α, β ⩽ 0.5 and 1.3 ⩽ λ ⩽ 2.9 - 2α, the Shannon entropy increases if λ increases; when 0.1 ⩽ α, β ⩽ 0.5 and λ ⩾ 2.9 - 2α, the Shannon entropy increases if λ decreases; when 0.5 ⩽ α, β ⩽ 0.9 and λ ⩽ 1.9, the Shannon entropy increases if λ increases; when 0.5 ⩽ α, β ⩽ 0.9 and λ ⩾ 1.9, the Shannon entropy increases if λ decreases.

The values of Shannon entropy are more than 1.774 in all the cases, which is far higher than that of the general model in Section 6. It shows that our model can better help investors disperse risk no matter how the parameters change.

In conclusion, assets A₂, A₃, A₄, A₈, A₁₀ are worth selecting and A₅, A₆ can be selected according to investors’ preference. And then, with a sufficient risk dispersion, investors can decide the values of parameters α, β and λ according to their risk preference. If investors are risk averse, the values of these parameters should be small; Otherwise, investors can choose a small value for λ and a large value for α, β.

7.2 The sensitivity analysis of investors’ tolerance level

In this part, the parameter δ in Model (32) is changed from 0 to 1 to analyze the relationship between optimal portfolio strategy and the investors’ preference for assets. The results are shown in Tables 10 11 and Figs. 6 7.

Table 10
The optimal portfolio strategy of Model (32) with α = β = 0.88 and λ = 2.25

Assets A ₂ A ₃ A ₄ A ₅ A ₈ A ₁₀

δ = 0 0.1667 0.1667 0.1667 0.1667 0.1667 0.1667

δ = 0.1 0.1667 0.1667 0.1667 0.1667 0.1667 0.1667

δ = 0.2 0.1667 0.1667 0.1667 0.1667 0.1667 0.1667

δ = 0.3 0.1667 0.1667 0.1667 0.1667 0.1667 0.1667

δ = 0.4 0.1667 0.1667 0.1667 0.1667 0.1667 0.1667

δ = 0.5 0.1667 0.1667 0.1667 0.1667 0.1667 0.1667

δ = 0.6 0.1691 0.1642 0.1691 0.1691 0.1642 0.1642

δ = 0.7 0.1886 0.1587 0.1886 0.1647 0.1467 0.1527

δ = 0.8 0.1908 0.1652 0.1807 0.1558 0.1408 0.1667

δ = 0.9 0.1957 0.1694 0.1717 0.1497 0.1374 0.1760

δ = 1 0.1997 0.1728 0.1645 0.1446 0.1351 0.1834

Assets	A ₂	A ₃	A ₄	A ₅	A ₈	A ₁₀
δ = 0	0.1667	0.1667	0.1667	0.1667	0.1667	0.1667
δ = 0.1	0.1667	0.1667	0.1667	0.1667	0.1667	0.1667
δ = 0.2	0.1667	0.1667	0.1667	0.1667	0.1667	0.1667
δ = 0.3	0.1667	0.1667	0.1667	0.1667	0.1667	0.1667
δ = 0.4	0.1667	0.1667	0.1667	0.1667	0.1667	0.1667
δ = 0.5	0.1667	0.1667	0.1667	0.1667	0.1667	0.1667
δ = 0.6	0.1691	0.1642	0.1691	0.1691	0.1642	0.1642
δ = 0.7	0.1886	0.1587	0.1886	0.1647	0.1467	0.1527
δ = 0.8	0.1908	0.1652	0.1807	0.1558	0.1408	0.1667
δ = 0.9	0.1957	0.1694	0.1717	0.1497	0.1374	0.1760
δ = 1	0.1997	0.1728	0.1645	0.1446	0.1351	0.1834

Table 11

The return, risk, Shannon entropy, θ and PCC of Model (32)

Assets	Return	Risk	Entropy	θ	PCC
δ = 0	0.1876	2.5121	1.7918	0.3775	0.5890
δ = 0.1	0.1876	2.5121	1.7918	0.3775	0.5890
δ = 0.2	0.1876	2.5121	1.7918	0.3775	0.5890
δ = 0.3	0.1876	2.5121	1.7918	0.3775	0.5890
δ = 0.4	0.1876	2.5121	1.7918	0.3775	0.5890
δ = 0.5	0.1876	2.5121	1.7918	0.3775	0.5890
δ = 0.6	0.1876	2.5129	1.7917	0.3770	0.6000
δ = 0.7	0.1890	2.5281	1.7869	0.3668	0.7000
δ = 0.8	0.1894	2.5467	1.7870	0.3544	0.8000
δ = 0.9	0.1898	2.5605	1.7854	0.3452	0.9000
δ = 1	0.1902	2.5712	1.7830	0.3381	1.0000

Fig. 6

The optimal portfolio strategy of Model (32) with α = β = 0.88 and λ = 2.25.

Fig. 7

The return, risk (/13), Shannon entropy (/10), θ(/2) of Model (32).

From Tables 10 11 and Figs. 6 7, we can find that:

When δ ⩽ 0.5, there is no change in the optimal portfolio strategy. The reason is that PCC = 0 .5890 ⩾ δ when δ ⩽ 0.5. In other words, Model (32) can provide investors with a strategy that satisfies at least 58% of their preference.

When δ ⩾ 0.5, the return and risk obtained by Model (32) increase if δ increases. However, the sensitivity of return to δ is less than that of risk, which makes the value of objective θ decrease. It means that if investors hold a lower tolerance level, the portfolio strategy will be worse, which is rational in reality: the actual optimal portfolio strategy always conflicts with investors’ irrational psychology.

The Shannon entropy of the strategy provided by Model (32) decreases slowly when δ increases. In fact, from Table 10 and Fig. 6, we can find that all the assets hold the same investment ratio when δ ⩽ 0.5. In this case, it is easy to prove that the Shannon entropy is at its maximum. When δ ⩾ 0.5, the balance is broken and the value of entropy decreases.

The investment ratio allocated to A₂ is the maximum at all times. In addition, the investment ratio allocated to A₂ increases with the increase of δ when δ ⩾ 0.5. Conversely, the investment ratio allocated to A₈ is the minimum and it decreases with the increase of δ when δ ⩾ 0.5. The reason is that the relative fuzzy prospect value of A₂ is the highest and that of A₈ is the lowest. Thus, if the investors’ tolerance level decreases, A₂ would be allocated more investment ratios and A₈ would be allocated fewer, which also demonstrates that the definition of prospect consistency coefficient is rational.

7.3 Comparison with previous studies

In order to better illustrate the advantages and limitations of our proposed model, we compared it with some existing studies:

(i) The Comparison with Zhou and Xu [45]:

In [45], Zhou and Xu proposed a novel model under the intuitionistic fuzzy environment. This model took the investors’ preference for risk into account but ignored the investors’ preference for assets. Comparatively, it seems that our model just focuses on the investors’ preference for assets. In fact, as mentioned in Section 7.1, if the investor is risk averse, they can select small values of α, β and λ to reduce the risk; Otherwise, they can choose a small value for λ and large values for α, β. Thus, our model also takes the investors’ preference for risk into account. Another difference between [45] and our model is that Zhou’s model can provide investors with suitable strategy when the historical data is unavailable, and our model needs sufficient historical data to quantify the return rates of assets.

(ii) The Comparison with Kar et al. [46]:

In [46], Kar proposed a bi-objective portfolio selection model based on fuzzy Sharpe ratio and fuzzy VaR ratio. This study adopted new return measure and risk measure, fuzzy Sharpe ratio and fuzzy VaR ratio, to evaluate the portfolio strategy. Comparatively, the possibilistic mean and variance used in our model are conventional. However, Kar’s model did not consider investors’ preference, whether for risk or assets, which may offer investors portfolio strategies that are inconsistent with their preference.

(iii) The Comparison with Sui et al. [47]:

In [47], Sui set up a possibilistic portfolio selection model with liquidity constraint. This model regarded asset liquidity as fuzzy variable, and then introduced the possibilistic liquidity constraint into portfolio selection model to help investors make better use of their capital. Similar to [45], it also took the investors’ preference for risk into account but ignored the investors’ preference for assets. In addition, from Tables 3 5 in [47], it could be found that Sui’s model may not allocate any investment to a selected asset. In contrast, according to Theorem 4, as long as δ > 0 (i.e. investors are not completely rational) in Model (31), our model can ensure that each asset will be allocated a certain amount of investment and the risk will be dispersed better.

8 Conclusion

For portfolio selection problem, considering that most studies have taken the investors’ preference for risk into account but ignored the investors’ preference for assets, in this paper, the prospect theory and possibility theory are combined to provide investors with a strategy that satisfies their preference for assets. Firstly, a novel reference point is proposed based on the joint possibility distribution of assets, which can provide investors with a comprehensive impression of these assets. Secondly, the prospect returns and risks of each asset are obtained by calculating the possibilistic mean and possibilistic variance of these assets. In addition, we defined the fuzzy prospect value of asset to denote the score of an asset in investors’ subjective cognition. For it can describe the prospect return rate of an asset under unit risk, it is utilized to rank the assets. On this basis, a prospect asset filtering frame is put forward. This framework can help investors solve the problem of how to select a certain number of assets from a large set. Especially, according to the rule of Selection Operation, it can ensure that all the selected assets have the highest scores, i.e. the highest fuzzy prospect values to satisfy investors’ preference.

In order to measure the consistency between a portfolio strategy and investors’ preference with range [0, 1], a new definition called prospect consistency coefficient is proposed. And then, considering that portfolio selection model may provide investors with strategies inconsistent with their preference for assets, we introduce the prospect consistency constraint with a parameter into the model. This parameter is used to describe investors’ tolerance level on the deviation of a portfolio strategy from their preference. Within the investors’ tolerance level, the model will provide investors with an optimal strategy to make profit and reduce risk as much as possible. Furthermore, it is pointed out that with the increase of investors’ tolerance level, the portfolio strategy provided by the model will be better, which indicates that the investors’ preference for assets is one of the irrational psychologies. Finally, a numerical example and the sensitivity analysis are given to show the efficiency of our proposed method. The results point out that compared with the general model, our model can not only better satisfy investors’ preference for assets, but also disperse risk effectively.

In the future, we will combine prospect theory with multi-period investment and then study the problem of multi-period investment based on investors’ preference for assets.

Footnotes

Acknowledgments

This research was supported by the “Humanities and Social Sciences Research and Planning Fund of the Ministry of Education of China, No. x2lxY9180090”, “Natural Science Foundation of Guangdong Province, No. 2019A1515011038”, “Soft Science of Guangdong Province, and No. 2018A070712006, 2019A101002118”, and “Fundamental Research Funds for the Central Universities of China, No. x2lxC2180170”. The authors are highly grateful to the referees and editor in-chief for their very helpful comments.

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Fuzzy portfolio selection with prospect consistency constraint based on possibility theory

Abstract

Keywords

1 Introduction

2 Related work

2.1 Portfolio selection based on probability theory

2.2 Portfolio selection based on fuzzy theory

2.3 Decision-Making based on prospect theory

3 Preliminaries

3.1 Possibility theory

4 A novel prospect asset filtering frame for portfolio selection

4.1 A novel reference point for comprehensive comparison

5.1 The prospect inconsistency coefficient and prospect consistency coefficient

Table 1 The return rates of assets A and B Asset Return 1 Probability 1 Return 2 Probability 2 A 30 0.5 0 0.5 B 10 0.5 0 0.5

6.1 Asset filtering process

7.1 The sensitivity analysis of parameters α, β and λ

8 Conclusion

Footnotes

Acknowledgments

References

Table 1
The return rates of assets A and B

Asset Return 1 Probability 1 Return 2 Probability 2

A 30 0.5 0 0.5

B 10 0.5 0 0.5